Question

what is the equation for this function? y = -cos x y = cos x y = sin x y = -sin x

Explanation:

Step1: Recall basic trigonometric graphs

The standard \( y = \sin x \) graph passes through \((0,0)\), rises to \(1\) at \(\frac{\pi}{2}\), etc. The standard \( y = \cos x \) graph passes through \((0,1)\), falls to \(-1\) at \(\pi\), etc.

Step2: Analyze the given graph

The graph in the image passes through \((0,0)\) (since it crosses the origin), which eliminates \( y = \cos x \) and \( y = -\cos x \) (as they pass through \((0,1)\) and \((0, -1)\) respectively). Now, check the direction: at \( x = 0 \), the graph is decreasing (going from \( y = 1 \) towards \( y = -1 \) as \( x \) increases from \( 0 \) initially? Wait, no, looking at the graph: when \( x = 0 \), the value is \( 0 \), and as \( x \) increases from \( 0 \), the graph goes down (negative direction). The standard \( y = \sin x \) at \( x = 0 \) is \( 0 \) and increases, while \( y = -\sin x \) at \( x = 0 \) is \( 0 \) and decreases. So the graph matches \( y = -\sin x \)? Wait, no, wait: Wait, the graph at \( x = 0 \) is at \( y = 0 \), and the shape: let's check key points. Wait, another approach: the graph of \( y = -\sin x \) is a reflection of \( y = \sin x \) over the x - axis. Let's check the starting point: at \( x = 0 \), \( y = -\sin(0)=0 \), and as \( x \) increases, \( y = -\sin x \) decreases (since \( \sin x \) increases from \( 0 \) to \( \frac{\pi}{2} \), so \( -\sin x \) decreases). Wait, but also, let's check the graph of \( y = -\sin x \): it has a zero at \( x = 0 \), \( \pi \), \( 2\pi \), etc. The given graph has zeros at \( 0 \), \( \pi \), \( 2\pi \), \( 3\pi \). Now, check the value at \( x=\frac{\pi}{2} \): for \( y = -\sin x \), \( y = -\sin(\frac{\pi}{2})=-1 \), which matches the graph (at \( \frac{\pi}{2} \), the graph is at \( -1 \)). For \( y = \sin x \), at \( \frac{\pi}{2} \), \( y = 1 \), which doesn't match. So the correct equation is \( y = -\sin x \)? Wait, no, wait the options: let's re - evaluate. Wait, the graph: when \( x = 0 \), the function value is \( 0 \), so it's a sine - type function (since cosine starts at \( \pm1 \) at \( x = 0 \)). Now, the direction: at \( x = 0 \), if we look at the graph, as \( x \) increases from \( 0 \), the function goes downward (negative y - direction). The standard \( y=\sin x \) at \( x = 0 \) has a positive slope (increases), while \( y = -\sin x \) has a negative slope (decreases) at \( x = 0 \). So the graph matches \( y = -\sin x \)? Wait, no, wait the options: wait, the graph in the image: let's check the key points. Wait, another way: the graph of \( y = -\sin x \) is equivalent to \( y=\sin(-x) \) (since \( \sin(-x)=-\sin x \)), which is a reflection over the y - axis? No, reflection over x - axis. Wait, let's check the value at \( x=\frac{\pi}{2} \): for \( y = -\sin x \), \( y=-1 \); for \( y = \sin x \), \( y = 1 \). The graph in the image at \( x=\frac{\pi}{2} \) is at \( y=-1 \), so that's \( y = -\sin x \)? Wait, but also, let's check the graph of \( y = -\sin x \): it passes through \( (0,0) \), \( (\frac{\pi}{2}, - 1) \), \( (\pi,0) \), \( (\frac{3\pi}{2},1) \), \( (2\pi,0) \), etc. The given graph seems to have those points. Wait, but the options: let's check again. Wait, the four options: \( y = -\cos x \) (starts at \( (0, - 1) \)), \( y=\cos x \) (starts at \( (0,1) \)), \( y=\sin x \) (starts at \( (0,0) \), increases), \( y = -\sin x \) (starts at \( (0,0) \), decreases). The graph in the image starts at \( (0,0) \) and decreases, so it's \( y = -\sin x \)? Wait, no, wait the graph in the picture: when \( x = 0 \), the y - value is \( 0 \), and the curve goes down first. So the correct function is \( y = -\sin x \)? Wait, but let's check the options again. Wait, maybe I made a mistake. Wait, the graph of \( y = -\sin x \): let's plot mentally. At \( x = 0 \), \( y = 0 \). At \( x=\frac{\pi}{2} \), \( y=-1 \). At \( x=\pi \), \( y = 0 \). At \( x=\frac{3\pi}{2} \), \( y = 1 \). At \( x = 2\pi \), \( y = 0 \). The given graph seems to have these points. So the correct answer is \( y = -\sin x \)? Wait, but the options are: \( y = -\cos x \), \( y=\cos x \), \( y=\sin x \), \( y = -\sin x \). So the correct one is \( y = -\sin x \)? Wait, no, wait the graph: wait, the initial point (leftmost) is at \( y = 1 \), then it goes down to \( y = 0 \) at \( x = 0 \), then down to \( y=-1 \), then up to \( y = 0 \) at \( x=\pi \), etc. Wait, maybe it's \( y = -\sin x \)? Wait, no, let's check the standard graphs again. The graph of \( y = -\sin x \) is a sine wave reflected over the x - axis. So the answer should be \( y = -\sin x \).

Answer:

D. \( y = -\sin x \)